Draw a shape on squared paper and plot a point to show its perimeter and area. Which points on the grid represent squares, rectangles, etc. Draw a shape that may be represented by the point (4, 12) or (12, 4). Find all the “impossible” points.

I like this problem a lot (I’ll spoil some of the fun in the comments) even though it’s fundamentally dissimilar to most of the problems I write about here.

One of the best parts about my working life right now, including grad school and my work with publishers, is my daily exposure to the vast set of answers to the question, “What makes for good math education?” That exposure has helped me find the edges of the usefulness of my own answer to that question, one which I’ve been developing for years, and for that I’m grateful. There’s nothing more pathetic than walking around convinced you’ve found the answer, forcing yourself to perceive other answers as either inferior to or derivative of your own, missing out on the bigness of the work of math education, on its richness and difficulty. Realizing the smallness of my own work relative to the whole has made me a much happier worker. That’s the odd thing.

So I’m grateful to instructors like Labaree and Stevens who urged us all to quit trying to solve the problem and focus first on describing the domain of the problem and its range of solutions. With that focus, I started to see fundamental similarities between this problem above and other problems I like.

They all use images to express their tasks concisely and to perplex the learner. That image alone would be enough to provoke some productive mathematical questions, even if none of them would necessarily be as productive initially as “Where are all the impossible points?” Perplexing images have that power. Also, imagine expressing that task using words alone. How much longer would that postpone the student’s encounter with the point of the task?

They all feature low threat levels and low barriers to entry. The task above allows the student to come up with a hypothesis and test it with new data instantly, from scratch, using nothing more than her mind, a piece of paper, and a pencil. The task allows a teacher to encounter a struggling student and say to her, “Would you just draw a rectangle for me? Any rectangle.” and start there. Once the student has graphed that rectangle on the plane the teacher can say, “Would you do that with five more rectangles and let me know what you notice, if anything?” The student can basically generate her own second act, which is better than most of the problems I design, which often require the advance knowledge of a certain mathematical model, without which you’re basically screwed.

They all ask you to understand the math forwards and backwards, inside and out. First it asks, “Given a shape, where’s the point?” Later it asks, “Given this point, what’s the shape?” This reversal of the question and the answer encourages students to understand their own thinking comprehensively.

2011 Oct 25: The action on Twitter indicates that few people are looking past the infographic itself before mashing the retweet button so let me put this out in front: I think it’s a losing game to share this image with math students.

Another day, another provider of online degrees looking to boost their PageRank by trading some trinket for a link. (Link responsibly.) In this case, we have an infographic from Rasumussen College. Click for larger.

A couple of quick ones on this:

I just don’t think you can ask a student to endure twelve years of frustrating math instruction now with the promise of a job making $70k as an architect later. It isn’t just kids who are lousy at delaying gratification like this, it’s everybody. And you’re asking them to do more than delay gratification. You’re asking them to delay gratification and embrace something they dislike. Tell me to put down the maple bar for the sake of a healthy heart later and I might — might! — accommodate you. Tell me to put down the maple bar and lick a cactus instead and I’ll definitely tell you where you can shove your infographic.

Good news about learning second-year algebra and trigonometry, though. It seems you’re well-positioned for a career teaching second-year algebra and trigonometry. (See also: careers in physics.)

A week ago, Jason Buell took on the tortured concept of the “real world,” particularly as it relates to due dates but also in the sense that everyone’s real world is different and immediate. I’m posting his last paragraph because it’s exactly right:

The other is that our students are living in the real world right now. There is nothing more real to a student than right now. Their friends, their enemies, their greatest loves and biggest heartbreaks, their passions, their hopes and their dreams are wrapped up in a few buildings, a quad area, and a blacktop. Saying this isn’t “the real world” diminishes everything there is about a student. Stop preparing kids for the real world and prepare them for right now.

And for the record, the differences here are real. Not real to 8th graders, but mathematically real. And actually some mathematically honest version of these differences is usually interesting to a class full of eighth graders …

There is a difference. The difference is important. Precision is important. Precision, for some students, is perplexing. No denying any of that. However, the attention paid that particular issue in that last thread was a dramatic instance of tree-noticing and forest-missing.

I’m trying to remember when I was 12 how I would react to We Use Math.

I was in New York last month consulting on a project intent on improving students’ perception of mathematics. We were spitballing around the table when someone pulled up this poster which was designed by NCTM:

Everyone in the room fawned over it, myself included. Then I pulled up this photo on my iPad.

I said I found the differences between the two images provocative. One person said, “Right. Basketball. Kids like basketball more than bridges.”

Is that it?

Let’s look at We Use Math, a project triple-teamed by Brigham Young University, the Mathematical Association of America, and the American Mathematical Society. Clearly that consortium brought a lot of resources to the table. The site features polished quotations from happy and well-paid professionals testifying to the usefulness of math in their careers. There’s a career tracker which lists dozens of high-paying careers, their salaries, their employers, the math required, and the ways math is used. Here’s the flyer for chemists [PDF], for instance. There’s stock photography of pretty people who (presumably) also use math, as well as a T-shirt store.

Engagement is a funny, fickle thing. On the subject of how to excite children about math in the same way it excites me, I have more questions than answers. Let me try to lay out a few markers, though:

There is a difference between showing a picture of the math of parabolas and provoking a question which can be answered using the math of parabolas.

When you see someone love something you find completely unloveable, it’s hard to relate to that person. It’s hard not to think they’re insane.

Venturing out farther on the creaky limb of engagement, here is some advice I give myself:

Don’t promise students they’ll enjoy the math they hate now in a career later. Let them experience math they enjoy now. PBS does this effectively with Get the Math. It features interviews with musicians, fashion designers, and video game engineers talking about how great math is, how much math they use, etc, but it also gives students something to do. It puts them in a position to experience that math now.

Show don’t tell. Instead of testifying to math’s power, show them math’s power. The CME Project features sidebars all throughout its textbooks with promises like:

(That’s setting aside the trickier issue of whether or not [x] interests students in the first place.)

An ounce of perplexity is worth a pound of engagement. Give me a student with a question in her head, one that math can help her answer, over a student who’s been engaged by a poster or a celebrity testimonial or the promise of a career. Engagement fades. Perplexity endures.

Perhaps it comes to this: rather than remembering your own tastes as a twelve-year-old, empathize with the tastes of a twelve-year-old who isn’t anything like you, one who has experienced only humiliation and failure in mathematics. What does math have to offer that student?

BTW: Here are three more PR projects. I like the odds on one of them way more than the other two.

Whatever interest attaches to your subject-matter must be evoked here and now; whatever powers you are strengthening in the pupil, must be exercised here and now; whatever possibilities of mental life your teaching should impart, must be exhibited here and now. That is the golden rule of education, and a very difficult rule to follow.

But another approach — since you used the word “enjoy” — is to simply consider math as an opportunity for puzzle-solving in interesting ways. After all, there is virtually NOTHING “personally relevant” in many of the games and pursuits people find so compelling, like, for example, Sudoku. Even chess. Or Angry Birds. Whether math is useful/relevant RIGHT NOW is a worthy and challenging goal.